## Section1.8Introduction Bootcamp

### Subsection1.8.6Miscellaneous

In Astronomy, a unit of distance, called parsec (pc), is in common use. It is defined to be the distance at which an object of size 1 Astronomical Units (AU) $(\approx 150\times10^6\ \text{km})$ will subtend an angle of 1 arc-second, which is equal to $1/3600$ of one degree. Two stars in a binary star system separated by $8.6 \times 10^{14}\ \text{m}$ are seen to subtend an angle of $0.2\ \text{arcsec}\text{.}$ How far away are the stars (a) in pc, and (b) in meters.

Hint

(a) $2.9 \times 10^4\text{ pc}\text{;}$ (b) $8.9 \times 10^{20}\text{ m}\text{.}$

Solution

A $5\text{-m}$ pole is attached to a boat. When the boat is $4\text{ km}$ away from the shore, you can see only $3\text{ m}$ of the pole above water. Use this information and assumption that Earth is a spherical ball to estimate the radius of earth.

The British physicist G. I. Taylor argued that the radius $R$ of a spherically symmetric nuclear explosion must depend on the energy $E\text{,}$ the initial density of air $D\text{,}$ and time $t$ since explosion. Using dimensional analysis, find a formula for the radius at time t after the explosion. [Challenging problem]

Hint

$R=CE^{1/5}D^{-1/5}t^{2/5}\text{,}$ where C is a dimensionless constant.
By dimensional analysis, find a formula for the oscillation frequency of a star of radius $R$ and density $D\text{.}$ Note that you will also need the dimensions of Newton's gravitational constant also, which is $[G_N]=[L]^3[T]^{-2}[M]^{-1}\text{.}$ [Challenging problem]